PMP BASED PRODUCTION MODELS-DEVELOPMENT AND INTEGRATION. Richard E. Howitt University of California, Davis, U.S.A

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PMP BASED PRODUCTION MODELS-DEVELOPMENT AND INTEGRATION Rchard E. Howtt Unversty of Calforna, Davs, U.S.A howtt@prmal.ucdavs.

1 PMP BASED PRODUCTION MODELS-DEVELOPMENT AND INTEGRATION Rchard E. Howtt Unversty of Calforna, Davs, U.S.A Paper prepared for presentaton at the PMP, Extensons and Alternatve Methods Organsed Sesson of the XIth EAAE Congress (European Assocaton of Agrcultural Economsts) The Future of Rural Europe n the Global Agr-Food System Copenhagen, Denmar, August 23-27, 2005 Copyrght 2005 by Rchard Howtt. All rghts reserved. Readers may mae verbatm copes of ths document for non-commercal purposes by any means, provded that ths copyrght notce appears on all such copes.

2 PMP BASED PRODUCTION MODELS-DEVELOPMENT AND INTEGRATION Rchard E. Howtt Unversty of Calforna, Davs PMP Ten Years Later It s now ten years snce the frst conceptual artcle on PMP was publshed (Howtt 1995), and the basc dea of usng dual calbraton values to mpute the mplct costs of agrcultural producton seems to be qute robust. Whle there have been many mprovements on the orgnal dea, there are few publshed crtcsms of the basc approach. One crtcsm termed the Hecele-Wolff crtque s dscussed later. The development of PMP approaches has been covered by the paper by Henry de Frahan, but can be characterzed as a gradual evoluton of methods that are able to match the mprovement n the estmaton methods and data resources avalable to modelers. The use of maxmum entropy methods has enabled optmzaton modelers to establsh a contnuum, between tradtonal optmzaton models and econometrc models. The advances have taen the form of a wder range of functonal forms, mproved nput shadow value estmaton, and more formal and explct use of pror nformaton. A Note on the Hecele-Wolff Crtque In ther paper n the European Revew of Agrcultural Economcs, Hecele & Wolff (2003) ntroduce ther estmaton paper wth a short revew and crtque of postve mathematcal programmng (PMP). They conclude that because the orgnal PMP specfcaton uses a constraned optmzaton model, usually lnear, to generate values for the bndng resource constrants they wll be nconsstent wth the quadratc form of the model that s correctly defned as the process generatng model. Hecele and Wolff conclude that:..the set of equatons (2) cannot be seen as unbased estmatng equatons and wll generally yeld nconsstent parameter estmates f the true data generatng process s correctly descrbed by the quadratc model The pont of ths note s to show that under the orgnal PMP specfcatons, the dual values derved from the lnear constraned model for a gven observaton set are numercally dentcal to the dual values that the quadratc model would generate for the same observatons. In short, we show that the orgnal PMP specfcaton (Howtt 1995) wll generate a set of resource shadow values that are consstent and unbased for calbraton to a sngle observaton. However, ths does not detract from the central pont of the Hecele & Wolff paper, namely the desrablty and effcency of the smultaneous estmaton of shadow values and parameters. Usng the notaton n Hecele & Wolff and usng ther partton nto preferable actvtes ( l p ) and margnal actvtes ( l m ), the Lagrangan for the constraned lnear problem can be wrtten as: 2

3 p p p m m m p p m m 0 (1) L= ( p c ) l + ( p c ) l + λ ( b A l A l ) + ρ (( l + ε) l) (2) L p p p p p p p p ( ) ( ) 0 p l = p c l λ A l ρ l = l therefore p p p p ρ = p c A λ (3) L m m m m m m ( ) ( ) 0 m l = p c l λ A l = l Snce m A = B B p c 1 m m (4) λ = ( ) Ths s exactly equaton (5) n Hecele and Wolff. Now, formulatng the quadratc problem and dervng the resource duals n the same manner usng the Kuhn-Tucer complementary slacness condton. The Lagrangan n an unparttoned form s: (5) L= pl dl 0.5 l Ql+ λ ( b Al) L (6) ( ) l = pl dl l Ql λ Al = 0 l therefore A λ = p d Q l 1 If ths expresson s premultpled by AQ and has the bndng resource constrant equaton substtuted n, then one obtans equaton (4) n Hecele and Wolff. Snce, at the calbrated optmum, the preferred land allocatons are wthn ε of the base acres so that p 0 l l, and usng the margnal cost calbraton equaton (2) from Hecele and Wolff: 0 (7) d + Ql = c+ ρ Substtutng ths nto the frst order condtons for the PMP problem we obtan: (8) A λ = p c ρ For comparson wth the LP model, ths expresson can be parttoned nto the sets of preferred and m margnal actvtes. Snce ρ = 0 ths yelds the same expresson for A λ as the calbrated LP model, namely: p p p (8) A p λ = p c ρ and (9) A λ = p c m m m Equatons (8) and (9) show that for the general elastcty based calbraton model and basc lnear quadratc PMP model n Howtt (1995) and Henry de Frahn et al (2005), the shadow values derved from the constraned lnear model are dentcal to those from the resultng quadratc PMP model. However, the central pont that Hecele and Wolff are concerned about s not the calbraton problem, but the problem of fttng a cost functon and estmatng the shadow values for a set of observed land allocatons. Applcaton of the two stage approach calbraton approach to a data set depends on what one assumes about the data avalable. Hecele and Wolff correctly defne the true underlyng model to be the quadratc model. They pont out that f standard accountng costs are used n the frst stage, the set of dual values assumng lnear accountng costs wll dffer from the dual values generated by the true quadratc model. It s clear that the changes n the crop allocatons between sample observatons wll 3

4 generate dfferent costs, and consequently the dual values for the quadratc model wll dffer from the frst stage LP model wth constant accountng costs. In the above stuaton the QP model has addtonal nformaton over the LP model, n that t assumes nowledge of the true quadratc cost functon, and thus how costs change wth changed land allocatons. If the LP data set ncluded the crop margnal producton cost for each observaton, then each LP run based on the observed data, ncludng the margnal cost wll have the same shadow value as the QP model. Snce the data set for both models wll contan observatons on the reported margnal costs, and not the true cost functon, the above approach seems to put the two stage model on the same emprcal bass as the combned estmaton approach. GME estmaton can then be drectly appled to the margnal cost condtons usng the sets of shadow values generated by the constraned model as data as n equatons (16) and (17) n Hecele (2005). Ths two stage approach may have operatonal advantages over the more elegant Hecele-Wolff approach, shown n Hecele s (2005) equatons (16) and (17) and (18), snce the resultng b-level optmzaton problem, wth the embedded complementarty condtons, may be hard to solve. Hecele brefly dscusses potental problems. A very smple emprcal example of the two stage estmaton s demonstrated n a Gams program avalable from the author. In the emprcal test, a base quadratc cost functon s generated from a sngle data pont and pror elastcty estmates. Ths cost functon s used wth a set of Monte-Carlo generated margnal revenues to optmze the QP problem for a set of optmum levels and shadow values. A calbrated LP model s then run for the same set of margnal revenues and resource constrants. If a constant accountng cost s used for the margnal crops the shadow values dffer from the mantaned model as correctly stated by Hecele and Wolff. However, f the cost nformaton n the LP model s updated by usng the cost functon from the mantaned model, the resultng LP shadow values are dentcal to the mantaned model results. The resultng set of LP shadow values are now used to estmate the cost functon. Usng a separate stage to generate a set of shadow values seems redundant, but may be more effcent operatonally than the one stage soluton. To conclude ths pont, the Hecele and Wolff approach s an effcent estmaton approach, where suffcent data sets are avalable. In those cases where data s mnmal, and calbraton has to be used, users of the two-step calbrated LP approach can be assured that the resultng dual values for sngle realzatons are consstent between the LP and PMP models. PMP-GME Producton Functon Models Over the past several years I have been weaned away from my past wor wth dual symmetrc formulatons of producton problems, and have concentrated on prmal producton functons for two man reasons, namely data relablty and process model nteracton. Clearly there s the same nformaton n a cost or producton functon approach to agrcultural models, however, n my experence, farmers n all countres have better nformaton on ther yelds and nput quanttes than they have on ther margnal costs. It s very lely that ths farmer preoccupaton wth yelds rather than costs wll lead to more precse responses to data questons. The producton functon approach explctly models polcy response at both extensve and ntensve margns of producton. In addton, the GME estmates of producton functons parameters have to satsfy the margnal producton frst order condtons for all nputs and the average product yeld condton for land. Ths latter condton s not usually used n tradtonal estmaton, but has the advantage of assurng that when the margnal condtons are ntegrated bac to yelds, they are consstent wth the data that the decson maers and farmers are most famlar wth. The general producton functon specfcaton has advantages where the envronmental outcomes from the use of dfferent nputs have dfferent socal mpacts. In addton, the producton functon approach uses crop and factor specfc pror estmates of factor demand elastctes to defne the maxmum entropy support values, thus has a wder range of pror nowledge than a sngle supply elastcty. 4

5 Hecele and Brtz (1999) propose an alternatve to the maxmum entropy (ME) formulaton orgnally proposed by Pars and Howtt. In ther approach, Hecele and Brtz mpose the curvature on the cost or producton functon by usng two constrants to defne the Cholesy decomposton. Ths enables them to defne the support values for the parameters of the Hessan drectly, rather than through the Cholesy decomposton. In turn, ths drect defnton of the supports enables them to be defned usng pror nowledge of the elastctes. Whle there are prors to help defne the support values for the parameters, the defnton of the support values for the error terms are more ad hoc. The recommendatons that have been put forward by varous authors for handlng ths ssue have vared wdely, and as yet, no clear consensus has emerged from the lterature about how to best to address ths problem. Many authors nvoe the 3 σ rule advanced by Puelshem (1994). Others recommend that error bounds are set as wdely as possble, an approach that s potentally msleadng and could ntroduce addtonal bas n the resultng GME estmates. Msang and Howtt (2005) propose an alternatve to ths approach by ntroducng a moment constrant nto the GME problem, to mnmze the nherent bas n GME estmates, and to remove the nfluence of the errors supports on the resultng estmates. A moment-constraned GME estmaton procedure, whle adherng closely to the propertes of classcal estmators under good data condtons and preservng un-basedness, s stll able to explot the desrable propertes of an entropy estmator, namely, ts clear advantage n cases where there exsts a hgh degree of collnearty n the data or even negatve degrees of freedom. Constranng the GME-PMP estmates to be unbased requres a smple summng constrant over each set of error components, and does not seem to alter the soluton of the GME problem. Gven the common occurrence of small sample bas and collnearty n producton data sets the addton of a moment constrant seems a sensble precauton. The PMP-GME producton functon models are readly understandable to research scentsts n assocated dscplnes. Somehow, showng a colleague a three dmensonal plot of a producton surface and explanng the alternatve trade offs greatly ads communcaton between dscplnes. Currently ths type of producton functon model s beng appled to several nterdscplnary research projects. Specfcally the research projects nclude: the ntertemporal analyss of rrgaton water use, the effect of global clmate change on croppng patterns, and the optmal use of precson farmng and cover crops on carbon sequestraton. Measurng the Informaton Gan from the Dsaggregaton of Producton Models The developments of entropy modelng methods over the past ten years has provded modelers wth the ablty to reconstruct models at any level of aggregaton from calbrated ndvdual farm models to large sample aggregate natonal models. Ths modelng ablty rases the queston of what the optmal level of aggregaton s for a gven data set and model purpose. Optmal aggregaton s the trade off between the heterogenety and the nose n the data. If the sample s drawn from a truly homogenous populaton, then all varaton s due to nose n the sample and there s no nformaton gan by dsaggregatng the model. However, ths stuaton s rare for agrcultural and envronmental models, where the nherent varablty n the resource and clmatc envronment ntroduces some heterogenety n the sample. A way of measurng the nformaton gans from dsaggregaton of multcrop systems s proposed by Howtt and Reynaud (2003). The measure s termed the Dsaggregaton Informatonal Gan (DIG) measure. The DIG measure s based on the Shannon (1948) measure of nformaton, and has the followng propertes. (1) It ncreases monotoncally wth the heterogenety of the dsaggregated sample. (2) The gan from dsaggregatng a unform set of samples s zero. (3) The DIG measure s nvarant to changes n the number of dsaggregated samples and the varablty of the aggregated sample. 5

6 If there are crop types and dsaggregated sets, the total number of observaton beng *, the cross-entropy between the aggregate observed land shares,, and the true dsaggregate land shares, y, s: y (10) CE = y ln y If an estmate of the crop n regon s ŷ shares,, and the true dsaggregate land shares,, s: ŷ, then the cross-entropy between the estmated land ˆ yˆ (11) CE = yˆ ln y As a polar case, assume that there s no nformaton at the dstrct level, and the estmaton merely allocates the aggregate land share dstrbuton y to each dstrct. The nformaton content for ths crude aggregated estmate s the CE measure defned above. Now assume that we obtan an estmator of regonal crop land allocatons ŷ. The CE ˆ measure s an aggregate measure, n term of entropy, of how far the dstrbutons of estmates y ŷ y y are from the true dstrbutons. Note that the estmate ncorporates both the nformaton gan from dsaggregaton, and the nformaton loss from bas as the sample sze s reduced. The Dsagregaton Informatonal Gan (DIG) from the dsaggregated estmaton s defned as: yˆ yˆ ln ˆ y CE (12) DIG = 1 = 1 y CE y ln y The DIG s a measure of the proporton of dstrct-level heterogenety that s recovered by the estmator. In the case of a perfect dsaggregated estmate where ŷ = y,, the DIG s equal to 1. In ths ŷ case, we recover all the heterogenety. In the case of full aggregaton, ŷ = y, and the DIG s equal to 0, and we recover no nformaton at the dstrct level. In all other cases, the DIG s between 0 and 1. The DIG measure ncreases as the dsaggregated estmates get closer to the true dstrct land use dstrbutons y. Currently, the optmal level of dsaggregaton for producton functon estmates s beng emprcally nvestgated for a small survey data set for northern Mexco crop producton. Now that a contnuum between robust econometrc estmates and dsaggregated calbraton models s avalable to polcy modelers, t seems reasonable that the level of aggregaton s justfed on a formal bass. Snce polcy modelng s not concerned wth testng fundamental hypotheses, t seems logcal to gnore the tyrrany of degrees of freedom and justfy the level of model aggregaton on the bass of nformaton added to the polcy queston. Intal Thoughts on PMP- CGE model Interacton Subscrbers to the Gams lst-server wll have notced the growng popularty of CGE models for polcy analyss. CGE models have many strengths that are derved from ther generalty. However, the general specfcaton nvarably means that the specfcaton of the agrcultural and envronmental sectors ŷ 6

7 have to be aggregated both by commodty and regon. Ths aggregaton lmts the ablty of CGE models to nteract wth physcal process models that are often hghly dsaggregated. An ongong area of wor s to use a dsaggregated PMP-GME model of agrcultural producton as a ln between physcal process models and regonal CGE models. In one study, the polcy queston s: How do changes n envronmental regulatons mpact the well beng of farmers of dfferent szes and regons? The locaton n northern Mexco has farms that range from large commercal operatons to subsstence farms where off-farm employment and remttances provde most of the household ncome. Whle the regonal models can ln envronmental polces to aggregate regonal mpacts on labor, nput costs and secondary effects, lned CGE models provde the endogenous reacton of nput and output prces to changes n regonal allocatons. There are advantages and dsadvantages to performng polcy analyss wth lned component models. The man dsadvantage s n the lac of smultanety n the prce response between the models. Currently, we rely on an teratve nterchange between the CGE and producton model that s transmtted through vectors of nput and output prces. Whle prce convergence occurs much of the tme, a general theory that ensures convergence does not seem to be avalable. However, such lnages enable nterestng polcy questons such as the followng to be analysed. What would be the net effect on the ncome of dfferent farm types n Northern Mexco of a smultaneous reducton n the energy subsdy for pumpng rrgaton water, and an openng of the corn maret to nternatonal pressures from the NAFTA trade agreement? The advantage of havng decoupled models s n model calbraton, estmaton and debuggng. Modelers who have been faced wth reconclng the results of a massvely connected model often react by breang the combned model nto ts component parts, so why not leave them that way? In addton, dfferent dscplnes often wor on dfferent spatal and temporal scales. Aggregaton and dsaggregaton methods enable model results to be consstently transferred between models electroncally, and thus wth fewer errors. Conclusons Ths short paper started wth the vew that agrcultural producton and envronmental modelng has moved to an nterestng stage where there s a formal contnuum between econometrc and calbrated optmzaton models. The development of maxmum entropy estmaton methods and better data sets has opened the potental for many developments. In ths paper I have revewed four topcs of personal nterest. I wll leave t to my colleagues to cover the many other mportant topcs that the development of agrcultural polcy models has ntroduced. References Hecele, T.(2005) Shadow Prces n PMP and Consequences for Calbraton and Estmaton of Programmng Models. Paper presented at the XIth EAAE Congress, Copenhagen, August Hecele, T. and W Brtz. (1999) Maxmum Entropy Specfcaton of PMP n CAPRI. CAPRI Worng Paper 99-08, Unversty of Bonn. Hecele, T. and H. Wolff Estmaton of constraned optmzaton models for agrcultural supply analyss based on generalzed maxmum entropy European Revew of Agrcultural Economcs. (2003) vol 30, pp Henry de Frahan, B. J Buysse, P Polome, B Fernagut, O. Harmgne, L Lauwers, G. Van Huylenbroec, and J. Van Mensel. (2005) Postve Mathematcal Programmng for Agrcultural and Envronmental Polcy Analyss: Revew and Practce forthcomng- Kluwer. Howtt. R.E. (1995) Postve Mathematcal Programmng Amercan Journal of Agrcultural Economcs. 77(2) pp

8 Howtt, R.E. (2005) Agrcultural and Envronmental Polcy Models: Calbraton, Optmzaton and Estmaton. Downloadable pfd draft at Howtt,R.E. and A Reynaud Spatal Dsaggregaton of Agrcultural Producton Data usng Maxmum Entropy European Journal of Agrcultural Economcs Vol 30 :3 pp1-29 Msang, S. and R. E. Howtt (2005) Usng Moment Constrants n GME Estmaton: A Methodologcal Note Worng Paper. Department of Agrcultural and Resource Economcs, Unversty of Calforna, Davs, USA. Pars Q and R.E.Howtt An Analyss of Ill-Posed Producton Problems Usng Maxmum Entropy Amercan Journal of Agrcultural Economcs, 80 February Puelshem, F. The 3-Sgma Rule Amercan Statstcan. 48 (May, 1994): Shannon, C.E. (1948) A Mathematcal Theory of Communcaton Bell System Techncal Journal, 27, pp , pp

LECTURE NOTES Duality Theory, Sensitivity Analysis, and Parametric Programming

LECTURE NOTES Duality Theory, Sensitivity Analysis, and Parametric Programming CEE 60 Davd Rosenberg p. LECTURE NOTES Dualty Theory, Senstvty Analyss, and Parametrc Programmng Learnng Objectves. Revew the prmal LP model formulaton 2. Formulate the Dual Problem of an LP problem (TUES)